Background on Interatomic Potentials

Atoms and molecules interact through a combination of attractive and repulsive forces. At short range, electron clouds repel each other, while at longer range, weak attractions arise from induced dipoles. One of the most widely used mathematical models capturing this behavior is the Lennard-Jones potential. Originally proposed in the 1920s by John Lennard-Jones, it has become a staple of molecular simulations in chemistry, physics, and materials science.

The Lennard-Jones Equation

The potential energy as a function of separation $r$ is expressed as

$V\left(r\right)=4\epsilon [{\frac{\sigma }{r}}^{12}-{\frac{\sigma }{r}}^6]$

where $\epsilon$ sets the depth of the potential well and $\sigma$ defines the distance at which the potential is zero. The $r^{-12}$ term represents short-range repulsion, while the $r^{-6}$ term models long-range attraction due to London dispersion forces. The resulting curve features a minimum at $r=\sqrt{2}\sigma$, capturing the equilibrium separation where forces balance.

Choosing Units

This calculator expects $\epsilon$ in electronvolts (eV) and $\sigma$ and $r$ in Angstroms (Å). Internally, the script converts values so that the final potential energy is also reported in electronvolts. Using consistent units is essential; mixing nanometers with Angstroms or joules with eV would lead to incorrect results. You can easily convert between unit systems if needed, but sticking to eV and Angstroms keeps the calculation straightforward.

Interpreting the Results

Positive potential energy indicates net repulsion, whereas negative values correspond to attraction. At very small separations, the $r^{-12}$ term dominates, leading to steep repulsion that prevents atoms from overlapping. As $r$ increases beyond $\sigma$, the attractive $r^{-6}$ component takes over, reaching a minimum at approximately 1.122σ. Farther out, the potential gradually approaches zero, indicating that the atoms no longer strongly influence each other.

Applications in Molecular Science

The Lennard-Jones model underpins many classical molecular dynamics simulations. By summing pairwise potentials between all atoms in a system, researchers can approximate how gases, liquids, and solids behave under various conditions. Although real materials often require more sophisticated potentials, the Lennard-Jones form provides a simple yet insightful starting point. It illuminates why noble gases condense at low temperatures, how surface tension arises, and how nanoparticles self-assemble.

Limitations and Extensions

While versatile, the Lennard-Jones potential does not capture directional bonding or electronic polarization. Molecules like water, with strong hydrogen bonds, need more elaborate potentials to reproduce their unique properties. Nevertheless, Lennard-Jones remains valuable for simple nonpolar species and as a building block in more complex force fields. Many simulation packages combine it with electrostatic terms or angle-dependent contributions to model real materials accurately.

Historical Perspective

Lennard-Jones developed his potential in an era when quantum mechanics was still young. He sought a tractable expression that matched emerging data on noble gas interactions. Over time, computational chemists refined the parameters for various atom pairs, enabling predictions of boiling points, viscosities, and crystal structures. Today, the Lennard-Jones potential stands as a testament to the power of simplified models in revealing fundamental physical behavior.

Conclusion

By entering values for ε, σ, and r, this calculator evaluates the Lennard-Jones potential energy between two atoms. Adjust the parameters to explore how the interaction varies with distance or compare different substances. Whether you are studying basic gas dynamics or building a molecular simulation, understanding this classic potential is a key step toward grasping the subtle forces that govern matter at the nanoscale.